Four identical firms give an HHI of exactly 2500.0 — "highly concentrated" by the US thresholds — and a numbers-equivalent of exactly 4.0000. For k equal firms the index is always 10000/k.
4 firms, total 100.0000
HHI = 2,500.0000
That is highly concentrated on the US thresholds — below 1,000 unconcentrated, 1,000 to 1,800 moderately, above 1,800 highly. The numbers-equivalent is 4.0000: this market behaves like that many equally-sized firms. CR4 is 100.00%. A merger of the top two would add exactly 1,250.0000 to the index — that delta, not the level, is what merger review screens on.
HHI
2,500.000
highly concentrated
Numbers-equivalent
4.0000
equally-sized firms this is worth
CR4
100.00%
CR8: 100.00%
Top-two merger delta
1,250.000
exactly 2·s₁·s₂
Each firm with its share and its contribution to the index
Firm
Size
Share
Contribution
Share of the index
Firm 1
25.0000
25.0000%
625.000
25.00%
Firm 2
25.0000
25.0000%
625.000
25.00%
Firm 3
25.0000
25.0000%
625.000
25.00%
Firm 4
25.0000
25.0000%
625.000
25.00%
Because shares are SQUARED, the largest firm dominates the index. The top firm here contributes 25.0% of the total while holding 25.0% of the market — which is the whole reason the HHI is preferred to a concentration ratio that treats every firm in the top four alike.
k equal firms give exactly 10000/k The merger delta is exactly 2·s₁·s₂ Everything depends on how the market is defined
What this tool shows
Four equal firms give an HHI of exactly 2500.0 and a numbers-equivalent of exactly 4.0000; ten give 1000.0 and 10.0000. For k equal firms the index is always 10,000/k, so its reciprocal recovers the firm count exactly — which is what makes “this market behaves like 1.92 equal firms” a statement rather than an analogy. The second identity is the one regulators use: merging the top two adds exactly 2·s₁·s₂ to the index.
The HHI on the 0–10,000 scale, with each firm’s squared contribution
The numbers-equivalent, 1/HHI, which recovers a firm count exactly
CR4 and CR8, so the two families of measure can be compared
The exact HHI increase a merger of the top two would cause
The US DOJ/FTC concentration bands, applied rather than described
Any input scale — shares, revenues or units, summing to anything
HHI and CR Numbers-equivalent Merger delta DOJ bands
The market definition decides the answer more than the arithmetic does.
Updated 13 September 2026 · Works in any browser, no installation
The Herfindahl-Hirschman index is the sum of the squared market shares, on a scale where 10,000 is a monopoly. Squaring is what makes it different from a concentration ratio: a firm with twice the share contributes four times as much. Its reciprocal is the “numbers-equivalent”, the count of equally-sized firms that would produce the same index — which for a market of k equal firms is exactly k.
At a glance
Formula shown
HHI = Σ sᵢ² × 10,000 with shares as fractions, or Σ (share in percent)². For k equal firms every share is 1/k, so HHI = k·(1/k²)·10,000 = 10,000/k and the numbers-equivalent 10,000/HHI is exactly k. A merger of firms i and j replaces sᵢ² + sⱼ² with (sᵢ+sⱼ)², so the increase is exactly 2·sᵢ·sⱼ — independent of every other firm in the market.
Scenario support
Merger review and antitrust screening, competition analysis for a regulator or a filing, supplier and customer concentration in credit and procurement risk, portfolio concentration, and any question of the form "how much of this market sits with how few players?".
Educational estimate
Planning support from the values you enter — not professional advice.
Squaring is the whole design
A concentration ratio adds the top few shares. The HHI squares every share and adds them all, and the difference is not cosmetic.
A firm with twice the share contributes four times as much. So the index responds to the SHAPE of the distribution, not just to how much the leaders hold between them.
Which is why CR4 cannot distinguish the tool’s first and third presets. Four equal firms and one firm with 70% against three with 10% both have a CR4 of 1.0000 — the top four are the whole market in each. Their HHIs are 2500.0 and 5200.0.
And why ten equal firms score 1000.0 with a CR4 of only 0.4000. The concentration ratio says the top four hold 40%; the index says the market behaves like exactly ten equal players.
The top firm typically dominates the index. In the one-dominant preset a firm with 70% of the market supplies 94.2% of the HHI, which is the concentration the measure exists to surface.
The numbers-equivalent turns the index into a firm count
“HHI 5200” is a number on an arbitrary scale. Its reciprocal is a count anyone can picture, and the mapping is exact rather than approximate.
10,000 ÷ HHI is the number of equally-sized firms giving the same index. Four equal firms return exactly 4.0000, ten return exactly 10.0000 — the identity holds by construction.
The one-dominant market returns 1.9231. Four firms exist; competitively it behaves like fewer than two. That sentence is what the index is for.
It is also the “effective number” from ecology and information theory, identical to the inverse Simpson index on the entropy page — the same quantity under a different name in a different field.
Which makes it the honest way to compare markets with different firm counts, since the raw index conflates “few firms” with “uneven firms” and the numbers-equivalent resolves both onto one scale.
The merger delta is exact, and independent of everyone else
Merger review screens on the CHANGE in HHI rather than its level, and that change has a closed form with a surprising property.
Merging two firms replaces s₁² + s₂² with (s₁+s₂)², so the increase is exactly 2·s₁·s₂. Everything else in the market cancels.
On the merger preset that is 2 × 0.30 × 0.25 × 10,000 = 1500.0, which the tool computes and which no other firm’s share can change.
US thresholds treat a delta above 100 in a market above 1,000 as warranting scrutiny, and above 200 in a market above 1,800 as presumptively anticompetitive. A 1500-point delta is far past either.
Two small firms merging barely moves it. Two 5% firms give a delta of 50 — which is why the rule catches consolidation at the top and ignores it at the bottom, by design.
The market definition matters more than the arithmetic
Every HHI is an HHI of a market someone defined, and the definition moves the number far more than any modelling choice does.
Narrow the market and concentration rises. “Premium organic yoghurt in London” and “dairy products in the UK” produce completely different indices from the same firms.
Geography does the same. A national HHI can look unconcentrated while every local market is a duopoly — which is the standard pattern in hospitals, supermarkets and broadband.
Imports and potential entrants are usually excluded and usually matter. A domestic HHI of 4,000 in a market open to imports is a different competitive situation from the same figure behind a tariff wall.
Which is why antitrust cases are argued over market definition rather than over the index. The arithmetic here is five lines; the definition is the case.
Against the Gini and the concentration ratio
Three measures of unevenness that emphasise different parts of the distribution, and picking by convenience is how they get misused.
CR4 and CR8 are simple and discontinuous. A firm just inside the top four counts fully and one just outside counts not at all, so a tiny change in ranking can move the measure a long way.
The Gini coefficient measures inequality, not concentration. A market of a thousand identical firms has a Gini of 0 and an HHI of 10 — perfectly equal and perfectly unconcentrated. A market of two identical firms also has a Gini of 0 and an HHI of 5,000.
That is the key difference: the Gini ignores how many players there are and the HHI does not. For competition questions the number of firms is the point.
The HHI is also most sensitive at the top where the Gini is most sensitive in the middle — opposite emphases, which is why they can rank two markets differently.
Reporting an HHI
Four things, and the first is the one that decides everything downstream.
State the market definition — product and geography. Without it the index is a number about an unspecified thing.
Give the delta as well as the level for any merger question. The thresholds are defined on both, and the delta is exactly 2·s₁·s₂.
Give the numbers-equivalent. It costs one line and turns an index on an arbitrary scale into a firm count.
And say what was excluded. Imports, fringe players below a size cutoff and vertically integrated capacity all change the shares, and their treatment is a judgement rather than a calculation.
Method. Shares are computed from whatever scale the input arrives on, so revenues, units and percentages all give the same index and the values need not sum to 100. Both central identities are exact by construction rather than approximated: k equal firms return 10,000/k and a numbers-equivalent of exactly k, and the top-two merger delta is computed as 2·s₁·s₂ directly rather than by recomputing the index on a merged market — the suite asserts those two agree, which is what establishes that every other firm’s share genuinely cancels. Per-firm contributions are retained so the table sums to the reported index rather than being derived from it. A negative size, an empty list and a total of zero all return no result. That engine is verified on every change against 100 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Gini CoefficientGini with the Lorenz curve, top and bottom shares and the Palma ratio — because two societies with a Gini of exactly 0.50 can have bottom-50% shares of 0% and 22%.
Shannon EntropyEntropy in bits, nats and bans with per-symbol contributions, efficiency and perplexity — and the demonstration that a sorted sequence and its shuffle give identical values to the last bit.
Market ShareYour share of the market, your share relative to the largest rival, and the revenue a target share would need.
PercentageSolve X% of Y, what percent X is of Y, reverse percentage, increase/decrease, discounts, and tax, tip, or commission.
RatioSimplify a ratio and share a total by it, with each term's fraction of the whole shown — because 3 : 2 means three fifths, not three halves.
Coefficient of VariationRelative standard deviation with the scale trap made visible — the same five temperatures give a CV of 22.6%, 41.6% or 3.0% depending on the unit, because only a true zero makes the ratio mean anything.
An educational tool, not legal or competition advice. An HHI is an index of a market someone defined, and narrowing the product or geographic definition raises it substantially — antitrust matters are argued over that definition rather than over this arithmetic. The concentration bands quoted are US agency guidance and differ between jurisdictions.
Published an HHI tool with each firm's squared contribution, the numbers-equivalent, CR4 and CR8, and the US agency concentration bands applied rather than described.
Established two exact identities rather than quoting them. k equal firms give an HHI of exactly 10000/k — verified for k from 1 to 25 — so the numbers-equivalent recovers the firm count exactly: 2500.0 and 4.0000, 1000.0 and 10.0000.
And a merger of the top two adds precisely 2·s₁·s₂ to the index, independent of every other firm. The suite checks that against recomputing the index on the actual merged market across 200 generated markets, which is what proves the cancellation rather than asserting it.
Showed why squaring matters: four equal firms and one firm with 70% against three with 10% both have a CR4 of 1.0000 and HHIs of 2500.0 and 5200.0. In the second the top firm supplies 94.2% of the index while holding 70% of the market.
Contrasted it with the Gini on the same page: a thousand identical firms have a Gini of 0 and an HHI of 10; two identical firms have a Gini of 0 and an HHI of 5,000. The Gini ignores how many players there are and the HHI does not, which is the whole difference for a competition question.
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